Proportional-Integral-Derivative (PID) Control

Article Goal

Explain PID Control algorithm for feedback control systems in robotics and automation

What is PID Control?

PID Control (Proportional-Integral-Derivative) is a feedback control algorithm that continuously calculates an error value as the difference between a desired setpoint and a measured process variable. The controller attempts to minimize this error by adjusting a control input based on proportional, integral, and derivative terms.

The key insight is that effective control requires responding to current error (proportional), accumulated past error (integral), and predicted future error (derivative). This three-term combination provides robust control for a wide variety of systems while remaining computationally simple and intuitive to tune.

PID Control: Written Walkthrough

Core Concepts

1. Proportional Control (P) - Responds proportionally to the current error magnitude
2. Integral Control (I) - Eliminates steady-state error by accumulating past errors
3. Derivative Control (D) - Provides predictive control by responding to error rate of change
4. Feedback Loop - Continuously measures output and adjusts input to minimize error

The Math

Error Calculation

The error signal is the difference between setpoint and process variable: \(e(t) = SP(t) - PV(t)\)

Where:

• \(e(t)\) is the error at time \(t\)
• \(SP(t)\) is the setpoint (desired value)
• \(PV(t)\) is the process variable (measured value)

PID Control Equation

The control output combines three terms: \(u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt}\)

Where:

• \(u(t)\) is the control output
• \(K_p\) is the proportional gain
• \(K_i\) is the integral gain
• \(K_d\) is the derivative gain

Discrete-Time Implementation

For digital systems, the continuous equation becomes: \(u[n] = K_p e[n] + K_i \sum_{k=0}^{n} e[k] \Delta t + K_d \frac{e[n] - e[n-1]}{\Delta t}\)

Where:

• \(n\) is the current sample
• \(\Delta t\) is the sampling period

Transfer Function

In the Laplace domain, the PID controller has the transfer function: \(C(s) = K_p + \frac{K_i}{s} + K_d s\)

Tuning Parameters

Each gain affects system behavior:

• \(K_p\): Higher values increase responsiveness but may cause overshoot
• \(K_i\): Higher values eliminate steady-state error faster but may cause oscillation
• \(K_d\): Higher values improve stability but amplify noise

Algorithm Steps

1. Initialize Parameters - Set initial values for \(K_p\), \(K_i\), \(K_d\) and reset integral sum
2. Measure Process Variable - Read current system output \(PV(t)\)
3. Calculate Error - Compute \(e(t) = SP(t) - PV(t)\)
4. Proportional Term - Calculate \(P = K_p \cdot e(t)\)
5. Integral Term - Update integral sum and calculate \(I = K_i \cdot \sum e(t) \Delta t\)
6. Derivative Term - Calculate \(D = K_d \cdot \frac{de(t)}{dt}\)
7. Control Output - Combine terms: \(u(t) = P + I + D\)
8. Apply Control - Send control signal \(u(t)\) to actuators
9. Wait - Wait for next sampling period \(\Delta t\)
10. Repeat - Return to step 2 until setpoint achieved or system stopped

C++ Implementation

pid_control.cpp